Isotropic filters in spatial frequency domain by a photonic crystal slab

ABSTRACT

An isotropic imaging filter is provided that includes a photonic crystal slab, where the photonic crystal slab includes a square lattice of air through holes, a dielectric constant, a thickness (d), a through hole radius (r), and a lattice constant (a), where the square lattice of air holes are separated according to a value of the lattice constant, where the thickness is configured according to d=M(a), where the through hole radii is configured according to r=N(a), where the thickness and the hole radii are configured to generate isotropic bands of guided resonances of an incident image.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional PatentApplication 62/585,816 filed Nov. 14, 2017, which is incorporated hereinby reference.

STATEMENT OF GOVERNMENT SPONSORED SUPPORT

This invention was made with Government support under contractFA9550-17-1-0002 awarded by the United States Air Force. The Governmenthas certain rights in the invention.

FIELD OF THE INVENTION

The invention generally relates to optical filters. More particularlythe current invention relates to a single photonic crystal slabisotropic image filter device operating in the wavevector domain.

BACKGROUND OF THE INVENTION

Filtering in the wavevector domain is widely used in image processing.Due to the two-dimensional nature of images, isotropic filters, wherethe responses depend on the magnitude but not the orientation of thewavevectors, are particularly useful. The most commonly used isotropicfilters are high-pass, low-pass, band-reject, and band-pass filters, alldefined in the wavevector domain.

In conventional Fourier optics, to achieve filtering in the wavevectordomain, one first obtains a Fourier transformation of an image on aFourier plane by passing the image through a lens. One then performsspatial filtering on the Fourier plane, followed by a Fouriertransformation again through a second lens. Such a technique requireslong propagation distance and therefore results in a bulky system.

In recent years, there has been significant progress in usingnanophotonic structures to develop a compact device for analog opticalcomputing. Specifically, many efforts have been made on achievingspatial differentiation of an incoming image, which in fact correspondsto a high-pass filter in the wavevector domain. While most initial workson nanophotonic structures have focused on demonstrating one-dimensionaldifferential operators, more recent works have shown thattwo-dimensional differential operators, including, in particular, theLaplace operator, which is a high-pass filter, can be achieved usingnanophotonic structures in either reflection or transmission.

What is needed is a compact, real-time and high-throughput imageprocessing device with reduced energy consumption and increased speed.

SUMMARY OF THE INVENTION

To address the needs in the art, an isotropic imaging filter is providedthat includes a photonic crystal slab, where the photonic crystal slabincludes a square lattice of air through holes, a dielectric constant, athickness (d), a through hole radius (r), and a lattice constant (a),where the square lattice of air holes are separated according to a valueof the lattice constant, where the thickness is configured according tod=M(a), where the through hole radii is configured according to r=N(a),where the thickness and the hole radii are configured to generateisotropic bands of guided resonances of an incident image.

According to one aspect of the invention, the guided resonances caninclude an isotropic high-pass filtered image, an isotropic low-passfiltered image, an isotropic band-reject filtered image, or an isotropicband-pass filtered image, where the isotropic low-pass filtered imageand the isotropic band-pass filtered image comprise a reflected incidentimage, where the isotropic high-pass filtered image and the isotropicband-reject filtered image include a transmitted incident image. In oneaspect, the invention further includes a beam splitter, where the beamsplitter is configured to separate the incident image from the reflectedincident image.

In another aspect of the invention, the photonic crystal slab includes adielectric material.

According to another aspect, the invention further includes a uniformdielectric slab with a thickness d_(s) disposed proximal to the photoniccrystal slab, wherein an air gap d_(g) between the uniform dielectricslab and the photonic crystal slab is configured according tod_(g)=T(a), where d_(s) is configured according to d_(s)=B(a), whered_(s) and the d_(g) are disposed to set a background transmission tounity. In one aspect, the uniform dielectric slab gap factor T has avalue that disposes the dielectric slab in a position to set abackground transmission to unity.

In another aspect of the invention, the hole radii factor N has a valuein a range of N≤0.5.

In a further aspect of the invention, the thickness factor M and theradii N are optimized together to establish an isotropic band structureof a guided resonance.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1C show (1A) a schematic drawing of the geometry of a singleslab device having a photonic crystal slab with a square lattice of airholes, (1B) a schematic drawing of the geometry of a single slab devicehaving a photonic crystal slab with a square lattice of air holesseparated from a uniform dielectric slab by an air gap. The slabs have adielectric constant ε=4.67. The geometry parameters are d=0.50a,r=0.11a, d_(s)=0.12a, and for FIG. 1B d_(g)=1.93a, where a is thelattice constant. The coordinate system is shown above the device. Thearrow (L) indicates the direction of the incident light. The electricfield directions of the S- and P-polarized light are also indicated.(1C) Geometry of the photonic crystal slab differentiator, which shows aphotonic crystal slab separated from a uniform dielectric slab by an airgap. For ϵ=12, the geometry parameters are: d=0.55a, r=0.111a,d_(s)=0.07a, d_(g)=0.21a. The plane above and below shows the input andoutput image respectively. The transmitted image (e.g. ring) through thedevice is the Laplacian of the incident image (e.g. disk) illuminated bya normally incident light with frequency ω₀=0.47656×2πc/a, according toembodiments of the current invention.

FIGS. 2A-2F show (2A), (2C), and (2D), nearly isotropic photonic bandstructure of the photonic crystal slab shown in FIGS. 1A-1B near thefrequency ω₀=0.77098×2π/a. (2A) Band dispersions along Γ-X and Γ-Mdirections. (2C) Constant frequency contours of the lower band. (2D)Constant frequency contours of the upper band. (2B) Scheme of multiplefiltering functions of the device. When the light frequency is onresonance at the normal incidence [ω=ω₀, labeled by dark arrows in (2B)and bottom horizontal lines in (2E), (2F)], the device realizes anisotropic high-pass filter (Laplacian) at transmission and low-passfilter at reflection. When the light frequency is detuned from theresonance at the normal incidence [ω=ω_(k), labeled by gray arrows in(2B) and top horizontal lines in (2E), (2F)], the device realizes anisotropic band-reject filter at transmission and band-pass filter atreflection. (2E) Transmittance |t| for S-polarized light as a functionof ω and |k| near ω₀=0.77098×2πc/a along a general wavevector direction(ϕ=14°). Due to the isotropic band structure, S light only excites theupper band, and the transmission spectra are almost identical along anywavevector direction ϕ. |t|=0 when ω=ω_(k). (2F) Reflectance |r| forS-polarized light as a function of ω and |k| near ω₀=0.77098×2πc/a alonga general wavevector direction (ϕ=14°). The reflection spectra areidentical along any wavevector direction. |r| 1 when ω=ω_(k). In all ofthe plots, frequency ω-ω₀ is in units of 10⁻⁴×2πc/a, while |k|, k_(x),and k_(y) are in units of 10⁻³×2π/a, according to the current invention.

FIGS. 3A-3F show isotropic high-pass filter (Laplacian). (3A)Transmittance for S-polarized light |t_(s)| k_(x), k_(y) at thefrequency ω₀=0.77098×2πc/a. (3B) |t_(s)| as a function of |k| along ageneral wavevector direction (ϕ=14°). (3C) Log plot of the Fouriertransform of the field profile for the incident image (3E): log|{tildeover (S)}_(in)|(k_(x), k_(y)). (3D) Log plot of the Fourier transform ofthe field profile for the reflected image (3F): log|{tilde over(S)}_(out)|(k_(x), k_(y)). The low wavevector components have beenfiltered out. (3E) Incident image |S_(in)|² of the Stanford emblem. Theimage size is 5220a×3456a. (3F) Calculated transmitted image |S_(out)|²,which shows the edges with different orientations. |k| k_(x), and k_(y)are in units of 10⁻²×2π/a, according to the current invention.

FIGS. 4A-4F show the isotropic low-pass filter performance. (4A)Reflectance for S-polarized light |r_(s)|(k_(x), k_(y)) at the frequencyω₀=0.77098×2πc/a. (4B) |r_(s)| as a function of |k| along a generalwavevector direction (ϕ=14°). (4C) Log plot of the Fourier transform ofthe field profile for the incident image (4E): log|{tilde over(S)}_(in)|(k_(x), k_(y)). (4D) Log plot of the Fourier transform of thefield profile for the reflected image (4F): log|{tilde over(S)}_(out)|(k_(x), k_(y)). The high wavevector components have beenfiltered out. (4E) Incident image |S_(out)|² of the Stanford emblemcorrupted by white noise. The image size is 5220a×3456a. (4F) Calculatedreflected image |S_(out)|², which reduces the white noise by imagesmoothing. |k| k_(x), and k_(y) are in units of 10⁻²×2π/a, according tothe current invention.

FIGS. 5A-5F show isotropic band-reject filter results. (5A)Transmittance for S-polarized light |k| k_(x), and k_(y) at thefrequency ω_(q)=0.77130×2πc/a. (5B) |t_(s)| as a function of |k| along ageneral wavevector direction (ϕ=14°). (5C) Log plot of the Fouriertransform of the field profile for the incident image (5E): log|{tildeover (S)}_(in)|(k_(x), k_(y)). The sinusoidal noise appears as peaks inthe spectrum, which lie on an approximate circle around the origin. (5D)Log plot of the Fourier transform of the field profile for thetransmitted image (5F): log|{tilde over (S)}_(out)|(k_(x), k_(y)). Thewavevector components corresponding to the periodic noise have beenfiltered out. (5E) Incident image |S_(in)|² of the Stanford emblemcorrupted by sinusoidal noise. The image size is 5220a×3456a. (5F)Calculated transmitted image |S_(out)|², which eliminates the periodicnoise. |k| k_(x), and k_(y) are in units of 10⁻²×2π/a, according to thecurrent invention.

FIGS. 6A-6F show isotropic band-pass filter results. (6A) Reflectancefor S-polarized light |r_(s)|k_(x), k_(y) at the frequencyω_(q)=0.77130×2πc/a. (6B) |r_(s)| as a function of |k| along a generalwavevector direction (ϕ=14°). (6C) Log plot of the Fourier transform ofthe field profile for the incident image (6E): log|{tilde over(S)}_(in)|(k_(x), k_(y)). The sinusoidal noise appears as impulses inthe spectrum, which lie on an approximate circle around the origin. (6D)Log plot of the Fourier transform of the field profile for the reflectedimage (6F): log|{tilde over (S)}_(out)|(k_(x), k_(y)). (6E) Incidentimage |S_(in)|² of the Stanford emblem corrupted by sinusoidal noise.The image size is 5220a×3456a. (6F) Calculated reflected image|S_(out)|² which isolates the periodic noise and simplifies itsanalysis. |k| k_(x), and k are in units of 10⁻²×2π/a, according to thecurrent invention.

DETAILED DESCRIPTION

Disclosed herein is a showing that several types of isotropic imagefilters in the wavevector domain can be implemented with a singlephotonic crystal slab device, according to the current invention.According to one embodiment, a slab is provided that is configured sothat the guided resonance near the Γ point exhibits an isotropic bandstructure. Depending on the light frequency and the choice oftransmission or reflection mode, the invention realizes isotropichigh-pass, low-pass, band-reject, and band-pass filtering in wavevectorspace. These filter functions are important for various image processingtasks, including edge detection, smoothing, white noise suppression, andsuppression or extraction of periodic noises. Further disclosed is anumerical demonstration of these filter functionalities by simulationsof a slab structure that is designed to operate in the visiblewavelength range. The current invention expands the application ofnanophotonics-based optical analog computing for image processing.

The purpose of filtering in the wavevector domain is to trans-form animage by modifying its Fourier transformation. In general, for anormally incident light beam along the z axis with a transverse fieldprofile S_(in)(x, y), the transmitted or reflected beam has a profileS_(out)(x, y)=

³¹ ¹[H(k_(x), k_(y)){tilde over (S)}_(in)(k_(x), k_(y))], where

⁻¹ is the inverse Fourier transform, {tilde over (S)}_(in)(k_(x), k_(y))is the Fourier transform of the input image, and H(k_(x), k_(y)) is thefilter transfer function. If H(k_(x), k_(y)=H(|k|) only depends on themagnitude of the wavevector |k|, the filter is isotropic. Here,k=(k_(x), k_(y)) refers to the in-plane wavevector.

According to one embodiment of the invention, the isotropic filters arerealized by a photonic crystal slab device, as shown in FIGS. 1A-1 c.The photonic crystal slab of the current invention exhibits a nontrivialisotropic band structure near the Γ point. The design details for such aphotonic crystal slab, see for example the embodiment shown FIG. 1C. Theexemplary embodiment shown in FIG. 1C can use either Si, GaAs, or anydielectric, where Si, or GaAs is applicable only in the infraredwavelength range. Further embodiments follow the same principle as theembodiment shown in FIG. 1C to enable a structure that can operate inthe visible wavelength range. In one embodiment, the dielectric constantof the material for the slabs is ε=4.67, which approximates that ofSi₃N₄ in the visible wavelength range. Provided here is a dielectricconstant that is real, as the extinction coefficient of Si₃N₄ at avisible wavelength is negligible. The photonic crystal slab has athickness of d=0.50a and a square lattice of air holes with radiir=0.11a, where a is the lattice constant. The thickness d and radius rare chosen to realize isotropic bands of guided resonances. A uniformdielectric slab with a thickness d_(s)=0.12a is placed in the vicinityof the photonic crystal slab. The air gap between the two slabs has athickness d_(g)=1.93a. d_(s) and d_(g) are chosen to set the backgroundtransmission to be unity. As a side note, the two-layer structurediscussed herein is quite compact. For a resonant wavelength λ=500 nm,a=650 nm, the total thickness is d+d_(g)+d_(s)=1.66 μm. Such a structurecan be fabricated using, e.g., focused ion beam assisted laserinterference lithography.

The photonic crystal slab in FIGS. 1A-1C hosts a pair of guidedresonances, which are degenerate at the Γ point with a frequencyω₀=0.77098×2πc/a. In general, in the vicinity of the Γ point, the bandstructure is highly anisotropic in the k space, assuming only the C_(4v)symmetry of the structure. However, remarkably, with the carefully tunedgeometry parameters above, both bands of the guided resonances exhibitalmost complete circular symmetry in the real part of theeigenfrequencies:ω_(k)≈ω₀ A|k| ²,  (1)where A₊=4.35, A⁻=−1.41 from fitting the band dispersion, and the upper(lower) sign corresponds to the upper (lower) band. In Eq. (1), andthroughout the rest of this disclosure, the wavevector will be in unitsof 2π/a.

The nearly isotropic photonic band structure (ω_(±)(k)−ω₀) for thestructures shown in FIGS. 1B-1C are plotted in FIG. 2A, FIG. 2C, andFIG. 2D. FIG. 2A shows that for both bands, the dispersions along theΓ-M and Γ-X directions have almost identical effective masses. Here, theeffective mass tensor is defined as m_(ij)*=ℏ[∂²ω(k)/∂k_(i)∂k_(j)]⁻¹.FIG. 2C and FIG. 2D show that the constant frequency contours for bothbands are almost circular. It is noted here that the radiativelinewidths γ_(±)(k), unlike ω_(±)(k) are anisotropic. Nonetheless,(γ_(±)(k)−γ₀) are much smaller than (ω_(±)(k)−ω₀), thus, they do notaffect the circular symmetry of the transfer functions much, as we willshow later.

In general, guided resonances in photonic crystal slabs may induce sharpFano resonance features in the transmission and reflection spectra. Forthe specific pair of guided resonances considered here, it has beenproved that, due to the isotropic band structure, S-polarized(P-polarized) light can only excite the upper (lower) band for everydirection of incidence. This effect is referred to as single-bandexcitation.

Due to the single-band excitation effect, as well as the presence of theuniform dielectric slab, which sets the background transmissioncoefficient to be unity, for this structure shown in FIGS. 1B-1C, thetransmission and reflection coefficients are

$\begin{matrix}{{{t_{\pm}\left( {\omega,k} \right)} = \frac{i\left\lbrack {\omega - {\omega_{\pm}(k)}} \right\rbrack}{{i\left\lbrack {\omega - {\omega_{\pm}(k)}} \right\rbrack} + {\gamma_{\pm}(k)}}},} & (2) \\{{{r_{\pm}\left( {\omega,k} \right)} = {{- e^{i\;\phi}}\frac{\gamma_{\pm}(k)}{{i\left\lbrack {\omega - {\omega_{\pm}(k)}} \right\rbrack} + {\gamma_{\pm}(k)}}}},} & (3)\end{matrix}$where the upper (lower) sign corresponds to S-polarized (P-polarized)light and upper (lower) band; ω is the incident light frequency.Therefore, on resonance,t _(±)(ω_(±)(k),k)=0, r _(±)(ω_(±)(k),k)=−e ^(iϕ),  (4)

The numerically determined transmission and reflection spectra forS-polarized light are plotted in FIG. 2E and FIG. 2F. Due to the effectof single-band excitation, S-polarized light only excites the upper bandof guided resonances. Moreover, as expected from Eqs. (1)-(3), theresultant transmission and reflection spectra are isotropic, i.e., thespectra are identical along any wavevector direction as defined by theazimuthal angle ϕ in FIGS. 1B-1C. When ω=ω(k), the transmittanceexhibits sharp dips with |t|=0, while the reflectance exhibits peakswith |r|=1, as expected from Eq. (4).

Depending on the operating conditions, the structures as shown in FIGS.1A-1C can be used to perform several very useful image processingfunctionalities.

Turning now to the isotropic high-pass filter embodiment, where theinventors have shown the k-dependent transmittance at the frequencyω=ω₀≡ω_(±)(k=0)

$\begin{matrix}{{{t_{\pm}\left( {\omega_{0},k} \right)}} = \frac{{{\omega_{\pm}(k)} - \omega_{0}}}{\sqrt{\left\lbrack {{\omega_{\pm}(k)} - \omega_{0}} \right\rbrack^{2} + {\gamma_{\pm}(k)}^{2}}}} & (5) \\{{{\approx \frac{{{\omega_{\pm}(k)} - \omega_{0}}}{\gamma_{0}}} = {\frac{A_{\pm}}{\gamma_{0}}{k}^{2}}},} & (6)\end{matrix}$

This transmittance realizes the Laplacian, a special isotropic high-passfilter.

Disclosed herein, it is show that the same device can provide a fewother very useful image processing functionalities under differentoperating conditions.

Regarding the isotropic low-pass filter embodiment, at the frequencyω=ω₀, if one considers instead the reflected light, the transferfunction is

$\begin{matrix}\begin{matrix}{{{r_{\pm}\left( {\omega_{0},k} \right)}} = \frac{1}{\sqrt{1 + {\left\lbrack {{\omega_{\pm}(k)} - \omega_{0}} \right\rbrack^{2}\text{/}{\gamma_{\pm}(k)}^{2}}}}} \\{\approx {\frac{1}{\sqrt{1 + {A_{\pm}^{2}{k}^{4}\text{/}\gamma_{0}^{2}}}}.}}\end{matrix} & (7)\end{matrix}$

This transfer function realizes an isotropic low-pass filter withreflection peak |r_(±)|=1 at the Γ point.

Regarding the isotropic band-reject filter, this embodiment isconfigured to operate away from the frequency ω₀, but at the frequencyω=ω_(±,q), where q is the amplitude of a non-zero in-plane wavevector,the transfer function then becomes

$\begin{matrix}{{{t_{\pm}\left( {\omega_{\pm {,q}},k} \right)}} = {\frac{{{\omega_{\pm}(k)} - \omega_{\pm {,q}}}}{\sqrt{\left\lbrack {{\omega \pm (k)} - \omega_{\pm {,q}}} \right\rbrack^{2} + {\gamma_{\pm}(k)}^{2}}}.}} & (8)\end{matrix}$

This transfer function realizes an isotropic band-reject filter withtransmission dip |t_(±)|=0 at |k|=q.

Turning now to the isotropic band-pass filter, at the frequencyω=ω_(±,q), the reflection has a transfer function

$\begin{matrix}{{{r_{\pm}\left( {\omega_{q},k} \right)}} = {\frac{1}{\sqrt{1 + {\left\lbrack {{\omega_{\pm}(k)} - \omega_{\pm {,q}}} \right\rbrack^{2}\text{/}{\gamma_{\pm}(k)}^{2}}}}.}} & (9)\end{matrix}$

This transfer function realizes an isotropic band-pass filter withreflection peak |r_(±)|=1 at |k|=q.

Therefore, devices according to the current invention achieve multiplefiltering functions. As schematically shown in FIG. 2B, when the lightfrequency is on resonance at normal incidence (ω=ω₀), the devicesoperate as an isotropic high-pass filter (Laplacian) at the transmissionmode and an isotropic low-pass filter at the reflection mode. When thelight frequency is detuned a bit from the resonance at normal incidence,the device operates as an isotropic band-reject filter at thetransmission mode and an isotropic band-pass filter at the reflectionmode, where the rejected or passed wavevector components are determinedby the light frequency detuning and polarization. For transmission mode,the transmitted image is the required filtered result. For reflectionmode, the reflected image is the required filtered result, which can beseparated from the incident image by using a beam splitter.

The filtering functions of the current invention are numericallydemonstrated herein. The isotropic high-pass filter (Laplacian) isdisclosed in FIG. 1C. Since the physical structure is different, a briefdiscussion of the performance of this device is disclosed herein as aLaplacian for completeness, with focus included on the other threeisotropic filters.

In all the numerical demonstrations below, the incident beam is Spolarized. The transmitted image is calculated following the standardway in image processing. (1) Compute the Fourier transform {tilde over(S)}_(in)(k_(x),k_(y)) of the incident field profile S_(in)(x, y). Notethe incident image is |S_(in)(x, y)|². (2) Compute the Fourier transformof the output field profile, {tilde over (S)}_(out)(k_(x),k_(y))=H(k_(x), k_(y)){tilde over (S)}_(in)(k_(x), k_(y)), whereH(k_(x), k_(y)) is the transfer function. (3) Obtain the output fieldprofile S_(out)(x, y) by inverse Fourier transform. Calculate the outputimage |S_(out)(x, y)|².

Regarding the isotropic high-pass filter (the Lapacian) FIG. 3 shows theisotropic high-pass filter (the Laplacian). FIG. 3A plots the filtertransfer function: the transmittance for S-polarized light|t_(s)|(k_(x), k_(y)) at the frequency ω₀=0.77098×2πc/a. The transferfunction is almost isotropic. FIG. 3B plots |t_(s)| as a function of |k|along a general wave-vector direction (ϕ=14°), and the fitting resultusing Eqs. (5) and (6), respectively. The fitting of Eq. (5) is almostperfect in the wavevector range as shown, while the quadratic fittingusing Eq. (6) is very good for |k| up to 0.6×10⁻²×2π/a. These plotsconfirm that the device indeed operates as an isotropic high-pass filter(the Laplacian) in this case. In FIG. 3A [and also in FIG. 4A, as shownlater], at larger wavevectors, the transfer function exhibits someanisotropy due to the dependency of the radiative linewidth γ(k) on thedirection of k.

The Laplacian enables image sharpening and edge detection. FIG. 3E showsan incident image of the Stanford emblem, while FIG. 3C plots theFourier transform of the field profile for this incident image. FIG. 3Dshows the calculated Fourier spectrum for the transmitted image, whichis obtained by a pointwise product of FIG. 3A and FIG. 3C. The lowwavevector components have been filtered out. FIG. 3F is the calculatedtransmitted image, which shows all the edges with differentorientations.

Turning now to the isotropic low-pass filter, FIG. 4 shows the isotropiclow-pass filter. FIG. 4A plots the filter transfer function: thereflectance for S-polarized light |r_(s)|(k_(x), k_(y)) at the frequencyω₀=0.77098×2πc/a. The transfer function is almost isotropic. FIG. 4Bplots |r_(s)| as a function of |k| along a general wavevector direction(ϕ=14°) and the fitting result of Eq. (7). The fitting is almost perfectin the wavevector range as shown. These plots confirm that the deviceindeed operates as an isotropic low-pass filter in this case.

The isotropic low-pass filter accomplishes image smoothing, withapplications ranging from character recognition in machine perception,preprocessing functions in the printing and publishing industry, tosatellite and aerial image processing. Here, one specific application ofthe low-pass filter is shown in white noise reduction. FIG. 4E shows anincident image of the Stanford emblem corrupted by white noise, whileFIG. 4C plots the Fourier transform of the field profile for thisincident image. FIG. 4D shows the calculated Fourier spectrum for thereflected image, which is obtained by a pointwise product of FIG. 4A andFIG. 4C. The high wavevector components have been filtered out. FIG. 4Fshows the calculated reflected image, where the white noise has indeedbeen reduced, demonstrating image smoothing.

Regarding the isotropic band-rejection filter, FIG. 5 illustrates theisotropic band-reject filter. FIG. 5A plots the filter transferfunction: the transmittance for S-polarized light |t_(s)|(k_(x), k_(y))at the frequency ω_(q)=0.77130×2πc/a. The transfer function is almostisotropic. FIG. 5B plots |t_(s)| as a function of |k| along a generalwavevector direction (ϕ=14°), which shows |t_(s)|=0 at|t|=0.84×10⁻²×2π/a. The transmittance curve is fit with Eq. (8) togetherwith Eq. (1) and γk≈γ₀. The fitting is almost perfect in the wavevectorrange as shown. These plots confirm that the device indeed operates asan isotropic band-reject filter in this case.

The isotropic band-reject filter can effectively eliminate periodicnoise, a common type of noise arising typically from electrical orelectromechanical interference during image acquisition. As periodicnoise appears as peaks in the Fourier transform at locationscorresponding to the wavevectors of the periodic interference, it can beisolated and filtered by band-reject filters. Here, shown is an exampleof periodic noise reduction with the isotropic band-reject filter. FIG.5E shows an incident image of the Stanford emblem corrupted bysinusoidal noise, while FIG. 5C plots the Fourier transform of the fieldprofile for this incident image. The periodic noise appears as spectralpeaks in the wavevector space, which lie on an approximate circle aroundthe origin. FIG. 5D shows the calculated Fourier spectrum for thetransmitted image, which is obtained by a pointwise product of FIG. 5Aand FIG. 5C. The spectral peaks corresponding to the periodic noise havebeen filtered out. FIG. 5F shows the calculated transmitted image, wherethe periodic noise has indeed been eliminated effectively. Noted here isthat the rejected wavevector where the maximum rejection occurs in ourband-reject filter can be easily tuned by tuning the light frequency.

Turning now to the isotropic band-pass filter, FIG. 6 illustrates theisotropic band-pass filter. FIG. 6A plots the filter transfer function:the reflectance for S-polarized light |r_(s)|(k_(x), k_(y)) at thefrequency ω_(q)=0.77130×2πc/a. The transfer function is almostisotropic. FIG. 6B plots |r_(s)| as a function of |k| along a generalwavevector direction (ϕ=14°), which shows |r_(s)|=1 at|k|=0.84×10⁻²×2π/a. The reflectance curve is fit with Eq. (9) togetherwith Eq. (1) and γ_(k)≈γ₀. The fitting is almost perfect in thewavevector range as shown. These plots confirm that the device indeedoperates as an isotropic band-pass filter in this case.

The isotropic band-pass filter performs the opposite operation of theband-reject filter. It is quite useful in isolating the effects on animage caused by selected wavevector bands. Shown here is an example ofextracting periodic noise patterns with the isotropic band-pass filter.FIG. 6E shows an incident image of the Stanford emblem corrupted bysinusoidal noise [same as FIG. 5E], while FIG. 6C plots the Fouriertransform of the field profile for this incident image. FIG. 6D showsthe calculated Fourier spectrum for the reflected image, which isobtained by a pointwise product of FIG. 6A and FIG. 6C. FIG. 6F showsthe calculated reflected image, where the periodic noise pattern isisolated and appears more clearly. This is useful because it simplifiesthe analysis of the noise, largely independent of the image content.

The design of isotropic wavevector domain image filters using a photoniccrystal slab is based on the guided resonances with isotropic bandstructure. The same idea can extend to other photonic structures thathost resonant modes with isotropic band structures. In particular, aphase-shifted Bragg grating can also perform the four filteringfunctionalities of our device, but with the transmission/reflection modeflipped.

The present invention has now been described in accordance with severalexemplary embodiments, which are intended to be illustrative in allaspects, rather than restrictive. Thus, the present invention is capableof many variations in detailed implementation, which may be derived fromthe description contained herein by a person of ordinary skill in theart. For example, one may use other photonic devices rather thanphotonic crystals that incorporate guided resonances with desired bandstructure, such as metasurfaces. One may use geometry that is differentfrom a square lattice of air holes; other lattices of other shapes ofholes are also possible. Moreover, our design can be readily extended tomultiple frequencies, by using a stack of multiple layers. Our designprinciple is not restricted to specific materials; it is general to anydielectrics.

All such variations are considered to be within the scope and spirit ofthe present invention as defined by the following claims and their legalequivalents.

What is claimed:
 1. An isotropic imaging filter comprising a photoniccrystal slab, wherein said photonic crystal slab comprises: i) a squarelattice of air through holes; ii) a dielectric constant; iii) athickness (d); iv) a through hole radius (r); and v) a lattice constant(a); wherein said square lattice of air holes are separated according toa value of said lattice constant, wherein said thickness is configuredaccording to d=M(a), wherein said through hole radii is configuredaccording to r=N(a), wherein said thickness and said hole radii areconfigured to generate isotropic bands of guided resonances of anincident image; further comprising a uniform dielectric slab with athickness d_(s) disposed proximal to said photonic crystal slab.
 2. Theisotropic imaging filter of claim 1, wherein said guided resonances areselected from the group consisting of an isotropic high-pass filteredimage, an isotropic low-pass filtered image, an isotropic band-rejectfiltered image, and an isotropic band-pass filtered image, wherein saidisotropic low-pass filtered image and said isotropic band-pass filteredimage comprise a reflected incident image, wherein said isotropichigh-pass filtered image and said isotropic band-reject filtered imagecomprise a transmitted incident image.
 3. The isotropic imaging filterof claim 1, wherein said photonic crystal slab comprises a dielectricmaterial.
 4. The isotropic imaging filter of claim 1, wherein an air gapd_(g) between said uniform dielectric slab and said photonic crystalslab is configured according to d_(g)=T(a), wherein said d_(s) isconfigured according to d_(s)=B(a), wherein said d_(s) and said d_(g)are disposed to set a background transmission to unity.
 5. The isotropicimaging filter of claim 4, wherein said uniform dielectric slab gapfactor T has a value that disposes said dielectric slab in a position toset a background transmission to unity.
 6. The isotropic imaging filterof claim 1, wherein said N(a) has a value in a range of N(a)≤0.5a. 7.The isotropic imaging filter of claim 1, wherein said M(a) and said N(a)are optimized together to establish an isotropic band structure of aguided resonance.